In mathematics, an **algebra homomorphism** is a homomorphism between two associative algebras. More precisely, if *A* and *B* are algebras over a field (or commutative ring) *K*, it is a function such that for all *k* in *K* and *x*, *y* in *A*,^{[1]}^{[2]}

The first two conditions say that *F* is a *K*-linear map (or *K*-module homomorphism if *K* is a commutative ring), and the last condition says that *F* is a (non-unital) ring homomorphism.

If *F* admits an inverse homomorphism, or equivalently if it is bijective, *F* is said to be an isomorphism between *A* and *B*.

## Unital algebra homomorphisms

If *A* and *B* are two unital algebras, then an algebra homomorphism is said to be *unital* if it maps the unity of *A* to the unity of *B*. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded.

A unital algebra homomorphism is a (unital) ring homomorphism.

## Examples

- Every ring is a -algebra since there always exists a unique homomorphism . See Associative algebra#Examples for the explanation.
- Any homomorphism of commutative rings gives the structure of a commutative R-algebra. Conversely, if S is a commutative R-algebra, the map is a homomorphism of commutative rings. It is straightforward to deduce that the overcategory of the commutative rings over R is the same as the category of commutative -algebras.
- If
*A*is a subalgebra of*B*, then for every invertible*b*in*B*the function that takes every*a*in*A*to*b*^{−1}*a**b*is an algebra homomorphism (in case , this is called an inner automorphism of*B*). If*A*is also simple and*B*is a central simple algebra, then every homomorphism from*A*to*B*is given in this way by some*b*in*B*; this is the Skolem–Noether theorem.

## See also

## References

**^**Dummit, David S.; Foote, Richard M. (2004).*Abstract Algebra*(3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.**^**Lang, Serge (2002).*Algebra*. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.